1993 AMC 12 Problem 17

Question

Amy painted a dart board over a square clock face using the “hour positions” as boundaries. [See figure.] If

t

is the area of one of the eight triangular regions such as that between

12

o’clock and

1

o’clock, and

q

is the area of one of the four corner quadrilaterals such as that between

1

o’clock and

2

o’clock, then

\frac qt=

2\sqrt3-2

\frac32

\frac{\sqrt5+1}{2}

\sqrt3

2

Scale the square so its sides are

x=\pm1

and

y=\pm1

The

1

o’clock ray meets the top side at

x=\frac{1}{\sqrt3}

Let the square be

[-1,1]^2

and its center be the origin. The

1

o’clock ray meets the top side at

(\frac{1}{\sqrt3},1),

so

t=\frac12\left(\frac1{\sqrt3}\right)(1)=\frac1{2\sqrt3}.

The corner quadrilateral has vertices

(0,0),

(\frac{1}{\sqrt3},1),

(1,1),

(1,\frac{1}{\sqrt3}).

Its area is

q=1-\frac1{\sqrt3}.

Hence

\frac qt=\frac{1-\frac{1}{\sqrt3}}{\frac{1}{2\sqrt3}}=2\sqrt3-2.

Thus the correct answer is A.

Problem 17 in other years